West Chester University
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Computer Science
College of Arts & Sciences
2014
An Infrastructure-less Vehicle Counting without
Disruption
Jie Wu
Temple University
Paul Sabatino
West Chester University of Pennsylvania
Jennifer Tsan
West Chester University of Pennsylvania
Zhen Jiang
West Chester University of Pennsylvania, zjiang@wcupa.edu
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Recommended Citation
Wu, J., Sabatino, P., Tsan, J., & Jiang, Z. (2014). An Infrastructure-less Vehicle Counting without Disruption. 43rd International
Conference on Parallel Processing, 520-528. http://dx.doi.org/10.1109/ICPP.2014.61
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more information, please contact wcressler@wcupa.edu.
An Infrastructure-less Vehicle Counting without Disruption
Jie Wu∗ , Paul Sabatino† , Jennifer Tsan† , and Zhen Jiang∗†
of Computer and Information Sciences, Temple University
† Dept. of Computer Science, West Chester University
jiewu@temple.edu, {PS772461, JT751802, zjiang}@wcupa.edu
∗ Dept.
Abstract—This paper presents a solution to count all moving
vehicles in a target region. This is a large-scale counting that
cannot be easily solved without a global view. However, there is
no single force that can provide such a global view. To achieve
an accurate result without either double- or miscounting,
the local counting at each checkpoint is synchronized in our
wireless communication by using the information carried by
vehicles along the traffic flow. Our analytical and experimental
results illustrate the correctness of the proposed scheme in both
closed and open road systems - even when the wireless signal is
affected by many factors. In this way, we provide an essential
support for the resource management in VANETs.
Keywords-Distributed algorithm, inter-vehicle wireless communication, resource management, traffic surveillance.
I. I NTRODUCTION
The automotive industry is about to enter a period of
transformative change, as relevant service support continues
to develop. Instantly counting all of a given type of moving
vehicles in the target region can provide critical information
for us to improve the current road services such as traffic
congestion control, disaster evacuation, and vehicle tracing.
Our target region can expand as large as an entire state, and
cannot have a single coverage at the global view level.
To precisely count the vehicles without any double- or
miscounting becomes a multi-site problem. Note that the
traffic flows dynamically, and each vehicle has an unpredictable speed, trajectory, and direction. It is very difficult for
any single checkpoint to determine its role in miscounting or
double-counting at the global view level. A synchronization
is needed among all checkpoints.
More importantly, we cannot rely on any traffic pattern,
which requires vehicles to change speed or trajectory in
order to achieve certain regularity. In the extreme case when
our counting scheme is applied to trace suspect vehicles,
the target can deliberately drive in an unpredictable manner
to avoid being caught, making any counting difficult to
implement. Second, any global information, such as VIN,
or centralized resource, such as the Internet access (of
DMV/NDR databases), can be too expensive to cover the
entire region, or becomes unavailable when it does not exist
or is damaged. The inconsistent local views among different
checkpoints increase the difficulty of our synchronization.
Our solution is motivated by the early work [1] to capture
a consistent global status (also called a “snapshot”) with
the distributed algorithm. In our approach, the counting is
initiated at the seed checkpoint(s) for all inbound traffic.
The active status (of site u), in one bit on/off information,
will be caught by the vehicles passing through via the
wireless communication. It will be carried to every adjacent
site along the traffic flow in a flooding-like manner. When
an idle site v received this information, it can start to
count its own inbound traffic. According to the order of
their initializations, u is the “predecessor” of v. The status
change at v will trigger a new flooding in the neighborhood.
When an activated site received the counting status of any
adjacent site, as the backwash of its outgoing flooding, it
can stop the local counting along such an inbound direction
without missing any vehicle. The entire process expands in
a cascading manner like a wave. By precisely controlling
the frontier wave and the backwash, our miscounting-free
approach will not have any double-counting.
In this paper, we synchronize the counting in the entire
target area under the “everyone” model, in which each site
will apply the same generic process in a fully-distributed
manner. The impact of the inconsistence among local views
can be mitigated. The counting converges with the ultimate
result at the global view level in a cost-effective way without any infrastructure support at the global view. The
contribution of our infrastructure-less counting is fourfold.
1) We first implement the counting scheme in an “everyone” model at each checkpoint in a fully-distributed
manner, where only short range surveillance and communication are available.
2) We prove that, by precisely controlling each checkpoint, active or not, in our synchronization, neither
mis- nor double-counting will occur.
3) Some extension work is provided so that our approach
will still be effective when the overtake, lossy communication, and odd traffic patterns are considered. We
also extend our work from the closed system to the
open road system. Moreover, we provide an efficient
method to gather those local views from the entire
system and then to constitute the global result.
4) We develop a simulation. Its results verify the correctness of our approach and show its scalability as a
practical solution for the large-scale problem.
The remainder of this paper is organized as follows:
Section 2 introduces the target problem and related work.
Section 3 provides some preliminary information. Section
4 presents our approach, extended from the closed system
under a simple road model to the open system under the
realistic road model. Its correctness will be analyzed and
then be verified in Section 5 with our experimental results.
Lastly, Section 6 concludes this paper and provides ideas for
future research.
II. TARGET PROBLEM
AND RELATED WORK
Under our “everyone” model, we catch every vehicle’s
coming in any direction at each intersection where we
install the camera accessories [2, 3]. To protect privacy, any
ownership information such as VIN cannot be accessible.
Considering the overhead cost and reliability of the entire
counting process, our counting process does not necessarily
rely on any centralized control or resource such as the Internet. Therefore, only exterior characteristics of the vehicle
such as color, brand, and type are used to identify the target
vehicle.
However, each local surveillance has a short range of
vision and cannot cover the entire road segment from one
end to the other. While the monitoring remains until each vehicle comes into the surveillance, some vehicles might have
traveled many sites and may have been counted multiple
times, i.e., double-counting. Unlike many existing services
(e.g., [4, 5]) that have independent job(s) at each site, the
statuses of each vehicle, counted or not, are interrelated
among different checkpoints. Because the traffic is often
unpredictable, the double-counting problem cannot be solved
completely by any deployment strategy of checkpoints.
Adopting image recognition to avoid double-counting is
costly and cannot ensure 100% accuracy, since surveillance
videos are taken from different angles and the vehicle’s
appearance is unpredictable.
In this paper, we provide a complete counting solution by
synchronizing all checkpoints with the information received
and carried by the vehicles. The recent technical advances of
wireless communication in VANETs can ensure the success
of each information exchange (e.g., [6]) and the delivery
in the expected direction (e.g., [7]). The relative location
of each adjacent vehicle in the flow can also be detected
by such communication among neighboring vehicles (e.g.,
[8]). Based on these, our counting will adapt to any dynamic
change along the road that is beyond the surveillance from
intersections. As a result, the counting is ensured 100%
correct.
III. P RELIMINARY
AND
N ETWORK M ODEL
Our counting is applied without any disruption to the
traffic. Table I summarizes all of the notations used in this
paper, which will be explained in the following.
Table I
N OTATIONS
u
{u, v}
u←v
u→v
ni (u)
no (u)
c(u, v)
c(u)
p(u)
s(u)
monitored intersection / checkpoint
a road segment with two adjacent intersections u and v
the inbound traffic of u along {u, v}
the outbound traffic of u along {u, v}
set of neighboring checkpoints of u along inbound traffic
set of neighboring checkpoints of u along outbound traffic
local counter of inbound traffic u ← v P
local counter of all inbound traffic at u, v∈ni (u) c(u, v)
the predecessor of u whose counting triggers the one at u
set of the successor(s) of u, i.e., no (u) − p(u)
A. Road
In this paper, we first adopt a simple road model, and
then extend the work to a more realistic road system. A
road segment, denoted by {u, v}, is a section of road that is
separated by two adjacent intersections u and v. Viewed
from the angle of u, outbound traffic from u to v and
inbound traffic from v to u are denoted by u → v and
u ← v, respectively. All adjacent intersections that can be
reached via the outbound traffic are denoted by no (u). All
adjacent intersections at the other end of each inbound traffic
flow are denoted by ni (u).
We assume that the road system is connected. To simplify
the discussion, we also assume that each road segment
is bidirectional (i.e., no (u) = ni (u) at each intersection
u) and there is no overtake allowed. Each time, only one
vehicle is allowed to enter the intersection and to make
the turn. In our extension, the real metropolitan map is
used and it allows multiple lanes and overtakes (i.e., traffic
is not always FIFO). Some of the road segments can be
one-way (i.e., unidirectional). That is, no 6= ni . Multiple
vehicles are allowed to pass the intersection simultaneously
and roundabouts are considered.
B. Vehicles
Each vehicle can change its speed and trajectory in an
unpredictable manner. It forms a node of VANETs, and its
built-in equipment has sufficient power and capabilities to
support the following functionalities:
• a directional communication [6] that can quickly send
and receive a short message,
• a coarse-grained collaboration based on the approach
in [8] that can detect whether this vehicle has been
overtaken by another, and
• a store of checkpoint status (also called the label, i.e.,
one-bit on/off information) and counting result that can
be carried to the next intersection and be forwarded to
the corresponding receiver.
C. Checkpoint, communication, and counting
A checkpoint will be set at every intersection. Any vehicle
entering the intersection u, say along inbound traffic u ← v,
will be identified by its exterior characteristics (e.g., color,
brand, or/and type) with a simple image recognition process
(e.g., [3]). It can be counted in c(u, v) before joining any
outbound traffic flow.
The checkpoint has the same communication and capability as a vehicle node of VANETs, in order to exchange and
share the information. In this way, u can obtain any status
update of an adjacent checkpoint v and its counting result
c(v) from the traffic u ← v.
By recursively accumulating information from each adjacent checkpoint, we can obtain the final result at the global
view level at a specific checkpoint, where the data sink is
available. This sink can also accept the signal to start the
entire counting process. Such a checkpoint is also called the
“seed.”
In our extensive work, we study the use of multiple seeds
in order to speed up our counting process. Additional work
is also provided under a more realistic model when the
communication is prone to failures by many factors.
IV. T HE P ROPOSED C OUNTING A PPROACH
Initially, each checkpoint is inactive. Our counting will
start from the seed checkpoint and at the corresponding
intersection. After each checkpoint has initiated, all the
vehicles passing through can be counted and will form the
flow spreading out along the outbound directions. When
any inactive checkpoint encounters such a counted flow, its
local counting will be activated and the above flooding will
continue. After all checkpoints become active, every vehicle
will have a chance to be counted, regardless of its driving
route.
Our proposed work focuses on the control at the frontier
of the propagation wave in order to avoid any doublecounting. We first show how all vehicles can be counted
precisely in the closed system under a simple road model
(see Alg. 1). Then we introduce a cost-effective method (see
Alg. 2) to collect the results that have been distributed to
each checkpoint, in order to form the global view at the seed.
After that, we will extend the above information constitution
and collection.
A. Our approach in a simple road model
The idea is demonstrated in Fig. 1. First, in the initial
phase (see Fig. 1 (a)), checkpoint 1 (which is equipped as
the only sink and seed) will initiate its local counting of each
inbound traffic, say 1 ← 2 and 1 ← 3. This is determined by
p(u) = φ and s(u) = no (u) = {2, 3}. Respectively, c(1, 2)
and c(1, 3) will be updated in phase 5 of Alg. 1.
In the simple road model, there is no overtake allowed,
and every information exchange between a vehicle and
the checkpoint is successful. After each local counting is
initiated, the vehicle first joining an outbound traffic will
always be upon the frontier wave of the flooding of counted
vehicles. With phase 2 of Alg. 1, such a vehicle, say along
Algorithm 1 Counting process, applicable to any intersection u under “everyone” model, in a closed, simple road
system
Require: Checkpoint u is deployed, with an initial “inactive” status, p(u) = φ, and s(u) = no (u).
Ensure: Each role of checkpoint u, the processing, and its
local view as the result of vehicle counting.
1: Initialization of an inactive seed checkpoint:
Activate local counting of each inbound traffic u ← v
where v ∈ ni (u).
2: Neighbor Synchronization from an active checkpoint:
Upon the frontier wave of counting, label the vehicle
first joining any outbound traffic to successors u → v ∈
s(u).
3: Propagation to an inactive non-seed checkpoint:
Set p(u) = {v} and s(u) = no (u)−p(u), when a labeled
vehicle enters along u ← v. Then activate the local
counting of each inbound traffic u ← w ∈ s(u).
4: Reaction of an active checkpoint under synchronization:
Stop the local counting in an inbound direction after a
labeled vehicle enters the intersection from that direction.
5: Counting of an active checkpoint:
Count in c(u, v), for any unlabeled vehicle entering via
the inbound
P traffic u ← v.
6: c(u) =
v∈ni (u) c(u, v) stabilizes after the local counting activated for each inbound direction has ended.
1 → 2 in Fig. 1 (a), will be notified with a label and then
will bring this counting status to checkpoint 2.
After the inactive checkpoint u = 2 observes such a
labeled vehicle approaching from v = 1, it can initiate its
local counting (see phase 3 of Alg. 1) in order to ensure
that each vehicle leaving from it has been counted. Using
the same way of the seed to label the outgoing vehicles, the
counting status will propagate until each checkpoint obtains
a chance to be active. According to the timing, we record
p(u) = {v} as the predecessor. Since all vehicles coming
along 2 ← 1 behind that labeled vehicle have been counted
by checkpoint 1, we initiate the local counting for 2 ← 3
only where ni (u) − p(u) = {3} (see Fig. 1 (b)).
As shown in Fig. 1 (b), when the active checkpoint u = 1
observes such a labeled vehicle approaching along 1 ← 2,
it will continue to count until that labeled vehicle enters the
intersection. Now, all uncounted vehicles that drive along
1 ← 2 can be counted by checkpoint 1. Since both 1 and 2
are in counting, no uncounted vehicle can enter 1 ← 2. We
stop the counting for 1 ← 2 (in phase 4 in Alg. 1) to avoid
any double-counting (see Fig. 1 (b)).
After receiving the label from each inbound traffic, 1 ← 2
in Fig. 1 (b) and 1 ← 3 in Fig. 1 (c), the activated local
c(1,2)
1
c(1,3)
1
1
c(2,1)
2
c(2,3)
(a)
c(1,2)
c(3,1)
2
3
1 c(1,3)
3
3
(e)
2
3
(b)
(c)
1
L (g)
2
c(2,3)
3
(f)
−1
L
2
c(3,2)
+1
c(2,1)
2
1
3
(d)
inactive state
active counting
counted traffic
L labeled vehicle
Algorithm 2 Information collection for the global view
where Alg. 1 is applied with a single seed checkpoint.
Require: Stable checkpoint u in Alg. 1 phase 6.
Ensure: Global view of vehicle counter.
1: If s(u) 6= φ, wait for c(v) (v ∈ s(u)) reported from the
vehicle coming along u ← v.
2: Ask a vehicle leaving along u → p(u) to carry the result
P
c(u) + v∈s(u) c(v) to p(u).
(h)
Figure 1. Counting process in the closed road system with 3 intersections,
where “1” is the seed and sink. (a) Initialization from seed(s), (b) propagation to a new checkpoint 2 and the corresponding synchronization, (c)
end of local counting at 1 and 2, (d) end of the entire counting at a global
snapshot, (e)-(f) possible intermediate configuration during the convergence
of counting process, and counting adjustment for (g) overtaking and (h)
being overtaken.
counting at checkpoint 1 can completely stop. The counters
c(1, 2) and c(1, 3) stabilize (in phase 6 of Alg. 1).
Finally, each checkpoint ends its counting. We have the
ultimate result, which has been distributed in the entire
region (see Fig. 1 (d)). In the following Theorem 1, we will
prove that this result is 100% correct, without any doubleor miscounting. However, the progress of the convergency at
each checkpoint is asynchronous; this is shown in Fig. 1 (c),
(e), and (f), where each presents a possible intermediate
configuration during the convergence from Fig. 1 (b) to (d).
The unpredictable nature of traffic flow causes the chaos of
the timing of each local counting and increases the difficulty
of our synchronization to avoid mis- or double-counting.
Theorem 1 (correctness in a closed system). Any misor double-counting can be avoided in the counting where
each checkpoint’s counting phase (i.e., phase 5 of Alg. 1) is
synchronized under our control in phases 1 to 4 of Alg. 1.
Proof: We first assume there is a miscounting. Note that a
seed checkpoint will apply phase 1 and a non-seed checkpoint will apply phase 3, in order to propagate their counting
status. In such a propagation, the successor set can include
every inactive adjacent checkpoint. The frontier wave will
form a closed line to include all activated checkpoints.
If this miscounted vehicle never passed though an active
checkpoint to experience the counting phase, it will always
travel outside of the counting area. This will lead to a
contradiction, because every inactive checkpoint will get
a chance to be active. Otherwise, this uncounted vehicle
must use the road segment of the predecessor to approach a
checkpoint in the counting area. Recursively, we can trace
back to the seed checkpoint along such a kind of inbound
traffic. This will lead to a contradiction of this uncounted
status, because the seed checkpoint will count all inbound
traffic. Therefore, we do not have any miscounting.
After that, we consider to avoid any double-counting.
When a counted vehicle enters an intersection along the road
segment from the predecessor, it will not be counted by the
setting of phase 3. Otherwise, it must use the traffic flow
from a successor. Then, it cannot surpass the labeled vehicle
going along the road segment from that successor (see phase
2), which will stop each counting process ahead (see phase
4). Therefore, the statement is proven.
Next, we focus on an effective method for the seed
checkpoint (also the sink) to collect stabilized counting
results from each site in the entire road system. An easy
way is to initiate a broadcast from each checkpoint to
the entire system. To reduce the complexity and overhead
cost, we simplify the collection process and present it in
Alg. 2. Basically, each checkpoint waits for the end of every
activated counting for the inbound traffic and obtains a stable
result at the local view level (phase 6 in Alg. 1). Then, along
the spanning tree built with p-s relation by phase 3 in Alg. 1,
each non-seed checkpoint u will accumulate the reports from
all successors (∈ s(u)) to its own, and then report this to the
predecessor (i.e., p(u)). The final report at the global view
level can be obtained at the root seed.
B. Extensive work
Extension to non-FIFO traffic model. In our extended road
model, the real metropolitan map is adopted. Multiple lanes
are used and each of them allows overtake. The collaboration
process in [8] is applied to detect the occurrence of overtake.
Note that we just need to confirm such an occurrence
of overtake before the labeled vehicle reappears in the
surveillance (of the next checkpoint), not exactly when,
where, or how long it took. After the labeled vehicle reaches
the other end of the road segment, the corresponding counter
needs to be adjusted as follows. For the labeled vehicle
overtaking an uncounted one, the latter one will miss our
counting (in either phase 3 or phase 4), and the counter
value must be corrected by an increase of one (i.e., +1 in
Fig. 1 (g)). Similarly, for each time the labeled vehicle is
overtaken by a counted one from behind, the counter must
be decreased by one (i.e., −1 in Fig. 1 (h)).
The vehicle-to-vehicle communication used in the labeling process is implemented by built-in equipment and cannot
be manipulated by the driver. Usually, the change of relative
location between vehicles is relatively slow, and gives us
sufficient time to achieve a reliable detection.
Extension to multi-target tracking. In this extended model,
we also allow multiple vehicles to enter the intersection
simultaneously and consider the surveillance of a possible
roundabout. With the existing techniques of image recognition (e.g., [3]), we can precisely identify each vehicle passing
through or parking around the intersection (or roundabout).
Extension to lossy communication model. In phase 2 of
Alg. 1, the label will be initiated, say at u, in order to
synchronize the counting at the adjacent checkpoint v. When
the information exchange with the outgoing vehicle fails,
the initialization of the counting at the inactive checkpoint
v will be delayed. This counted vehicle will be doublecounted as the flow enters the surveillance of any other
active checkpoint. Similarly, when checkpoint v has been
activated, the counting of v ← u must stop upon receiving
such a label. The delay will also incur double-counting.
To solve the problem, the local counter will be adjusted
(decreased by 1) until receiving can be confirmed with the
TCP acknowledgment in [6].
Extension for counting along one-way streets. One-way
street is designed to direct vehicles to move in one direction,
say from u to v. Thus, we have no (v) = ni (u) = φ. It
typically results in higher traffic flow as, we don’t have
to monitor the on-coming traffic in the other direction.
But synchronizing checkpoints at both ends u and v often
means taking extra work because the information delivery
is unidirectional. Alg. 3 shows a complete counting solution
with the above extensive considerations. Theorem 2 proves
that such counting is 100% correct, without any mis- or
double-counting.
Theorem 2 (correctness of extended counting scheme).
The counting with Alg. 3 can avoid any mis- or doublecounting.
Proof: We prove that Alg. 1 can directly be applied to oneway streets. The necessity and sufficiency of other additional
parts in Alg. 3 can be seen in the above discussion.
Any outbound traffic from u, if it is used in the
predecessor-and-successor relation and initiates the counting
at the adjacent checkpoint v, will not be counted at v
(because u = p(v) 6∈ s(u) in phase 3). That is, there is
no need for the labeling process along the opposite inbound
direction. Otherwise, v will be initiated earlier than u. The
labeling process at u (in phase 2) will end the counting at v.
This is correct since both u and v are active and there is no
uncounted vehicle traveling between them. From the view
of both u and v, Alg. 1 does not need to change. Thus, the
statement can be proven.
Extension for odd traffic pattern. In the above road system,
we assume that each counting initiated for an inbound
traffic u ← v will always have a chance to encounter
another vehicle carrying a label from v so that the result
can stabilize. When our counting is applied to a real road
system without such an assumption, we may have a deadlock
problem as addressed in [9]. If all vehicles deliberately
Algorithm 3 Constitution of local views in the closed
system, where overtake, multiple lanes, lossy wireless connection, and one-way streets all are considered.
Require: The same in Alg. 1.
Ensure: Local view of vehicle counter.
1: for each checkpoint u deployed do
2:
Apply Alg. 1.
3:
In phase 2, set c(u) = c(u) − 1 when u misses
the contact with the target vehicle in labeling and
cannot accomplish the notification process. Then the
checkpoint will repeat this trial of labeling process
in that direction until the first vehicle confirms the
receiving of label.
4:
Phase 5 counting is extended to a multi-target tracking.
5: for each labeled vehicle moving along u ← v do
6:
Apply collaborative V2V communication to maintain
its relative position to each moving vehicle nearby.
7:
When it overtakes an uncounted vehicle in the front,
c(u) = c(u) + 1.
8:
When it is overtaken by a counted vehicle from
behind, c(u) = c(u) − 1.
detour around and avoid entering a road segment in the
active status of counting, this counting cannot converge.
The corresponding directional road segment is called the
“orphan.” Such a starvation status also suspends the convergence of the adjacent checkpoints along the spanning
tree that is built with the predecessor-and-successor relation,
forming a waiting chain. We resort to police patrol because
the trajectory of vehicles is unpredictable and cannot be used
to build a reliable deadlock-free solution. In our approach,
every patrol car can store the on/off statuses of inbound
traffic counters of each checkpoint, and can have the same
ability as the counted vehicles to share information with a
checkpoint via wireless communication.
Theorem 3 (guaranty of the convergence). The counting
with Alg. 3 will converge and not have the deadlock problem
when any two adjacent checkpoints can be reached by a
patrol car in a finite delay τ < ∞.
Proof: Without loss of generality, we have two adjacent
checkpoints u and v (v ∈ s(u)). After v is visited, u will
eventually encounter such a patrol car and fetch the counting
status of v. According to the protocol in phase 4 of Alg. 1,
the counting of u ← v stops. Thus, the statement is proven.
Based on the above theorem, we design a cycle to cover
every checkpoint at least once. The following theorem ensures the existence of such a cycle in our closed road system.
In the ideal case, a Hamilton cycle can be used. Otherwise,
some checkpoints can be visited multiple times. Every police
car will evenly be distributed and drive along such a cycle.
Algorithm 4 Information collection for the global view
where Alg. 3 is applied.
Require: Counting at checkpoint u, stable with Alg. 3.
Ensure: Global view of vehicle counter.
1: Apply phase 1 in Alg. 2.
2: For each one-way traffic u → v in the above phase,
v ∈ s(u) ∧ v 6∈ ni (u), wait for c(v) reported from the
patrol vehicle coming along a circuitous route from v.
3: Apply phase 2 in Alg. 2.
4: For each one-way traffic u ← p(u) in the above phase,
p(u) 6∈ no (u), ask every patrol
P vehicle passing through
u to carry the result c(u) + v∈s(u) c(v) to p(u).
The patrol car will not be counted by any checkpoint, but
will help to deliver the “stop” signal to adjacent checkpoints.
As a result, each checkpoint will eventually stop its counting.
Theorem 4 (success ensured). In a closed road system with
directional road segments where each intersection can be
visited by vehicle(s) at least once, we can always find a
patrol cycle, but not necessarily a Hamiltonian cycle.
Proof: A closed road system is a connected network where
all vehicle trajectories can be connected as well. Since each
intersection can be visited at least once, the patrol path
can reach each intersection. Since the closed system does
not have any dead end and the entire region is limited, the
patrol path can always move forward and eventually reach an
intersection that was previously visited. Thus, the statement
is proven.
In Alg. 2, the stabilized counting result will be collected
along the traffic back to the predecessor. A one-way road
segment will force such a process to take a circuitous route,
which adds distance and time. To ensure the success of
information collection, we also rely to the police patrol. The
chance of using one-way road segment in the predecessorand-successor relation is so little. Moreover, in the real
world, many one-way streets have been upgraded to bidirectional [10]. So, our counting is practical to implement.
The revised information collection can be seen in Alg. 4.
Does anyone see that white van? − Extension for
counting a specified type. When the surveillance at each
intersection can identify certain kinds of vehicles from their
exterior characteristics, our counting can provide a complete
search for many applications. For instance, the Beltway
sniper-attacks took place over the course of three weeks in
October 2002 in Washington, D.C., Maryland, and Virginia
[11]. Early tips from eyewitnesses included reports of a
white box truck with dark lettering. Police across the area
and the state of Maryland were pulling over white vans and
trucks. Later, evidence shows that had this vehicle search
for a “white van” resulted in the shooter being caught, more
lives would have been saved.
We do not need to worry about information privacy. No
Algorithm 5 Constitution of local views in the open system.
Require: The same in Alg. 1.
Ensure: a complete status in the global view level.
1: for each checkpoint u deployed do
2:
Apply Alg. 3.
3:
In phase 3 of Alg. 3, the counting is activated for
not only inbound traffic but also outbound interaction
(i.e., −1 in counting phase 5).
4:
In phase 6 of Alg. 3, the result stabilizes after the
counting activated for each non-interaction (inbound)
traffic ended.
ownership information is used. Vehicle registration information provided by manufacturers in the standard can be used
to identify the uncounted vehicle in the type that we are
counting.
Extension with multiple seeds. In the above counting, the
entire region is equipped with one sink only. When multiple
sinks are equipped, a spanning tree of predecessor-andsuccessor relation will spread out from each sink. When
all trees use the same label, a forest will form, making the
tree-depth relatively shorter. This can quicken our counting
process.
Extension to the open road system. Then, we extend our
counting from a closed road system to an open system where
there are vehicles in and out along the border continuously.
A complete counting in the open system can be defined in
Def. 1, as follows.
Definition 1: A complete counting in an open road system
is a scheme that can count every vehicle inside the area
enclosed by the border, and everyone that travels in and
out along the border in any possibility. Such a global status
reached in the counting is called a “complete status.”
Definition 2: A border of an open road system consists of
all intersections that have both traffic flows in and out of the
system, either inbound or outbound. Specifically, the traffic
that starts/ends at an intersection along the border and is
connecting the outside of the system is called “interaction.”
Our strategy is to activate the counting of interaction
along the border (i.e., the surveillance of in-out traffic) while
accomplishing the counting inside the area that is enclosed
by the border line. As a result, our counting can quickly
reach a “complete status,” as we proved in Corollary 1.
Simply, we apply Alg. 3 to non-interaction traffic. We will
initiate each local counting for interaction, either inbound or
outbound, as well as other non-interaction inbound traffic at
the same checkpoint. Note that those counting processes on
the interaction will remain active for any possible vehicle
coming in or out at any time. The details can be seen in
Alg. 5.
Corollary 1 (convergence in open system). Initiated from
seed(s), Alg. 5 applied in the open road system can eventually reach the complete status.
Proof: The counting with Alg. 5 will propagate to every
checkpoint, as would the one with Alg. 3. The counting
along the border for interaction will remain active for any
possible vehicle that suddenly comes in or leaves. Based on
Theorem 3, other counting processes will stabilize due to
the update in phase 6. After that, based on Theorem 2, any
vehicle in a non-interaction traffic will eventually encounter
the frontier wave and be counted before the counting result
stabilizes. Any vehicle newly coming or leaving the system
can be observed by the active counting for interaction along
the border. Thus, the statement is proven.
Corollary 2 (correctness in asynchronous global view).
During the convergence of Alg. 5, there is no mis- or doublecounting at any active checkpoint.
Proof: Our counting is initiated from a limited number
of seeds. Before the counting process reaches the final
“complete status,” a vehicle can escape from any inactive
checkpoint along the border. If this vehicle has been counted,
it should surpass a labeled vehicle and has updated the
information (line 8 in Alg. 3). For any vehicle coming into
the counting area via such an inactive checkpoint, it will
eventually encounter the frontier wave of our counting and
have the information up to date. The rest of the proof is
referred to Theorem 2.
V. S IMULATION
We adopt the city map of Manhattan in New York
(from OpenStreetMap [12]), which contains many one-way
streets. The test focuses on the traffic from Central Park
to Madison Square Park in midtown. With different traffic
volumes changing from 10% to 100% of the average, the
trace data of each vehicle is generated by the simulation
SUMO [13]. Multiple lanes and overtakes are considered.
The instantaneous velocity is limited to 15 mph. We simulate
the surveillance at each intersection (or roundabout) and
lossy wireless communication, with a 30% chance of failure.
We also test with different numbers of seeds, ranging from
1 to 10, which is randomly selected from the available
checkpoints in the system.
We first close the traffic lanes along the border and build a
closed system. The checkpoint will repeatedly apply Alg. 3
and Alg. 4, playing its deterministic role locally under our
real road model of New York City. Initiated from the seeds,
our counting is expected to converge with the stable local
views. These results will be collected by the seeds as the data
sinks. We verified the correctness of our counting without
any mis- or double-counting. We study the performance
of information constitution and collection, in terms of the
elapsed time.
Fig. 2 shows the time needed for each checkpoint to
constitute a stable result with Alg. 3. Similarly, Fig 3 shows
the time of the whole counting until the data sinks fetch the
information with Alg. 4 and form the global view. Fig 4 (c)
shows the average time for Alg. 3 to converge when the
vehicle velocity limit is lifted from 15 mph to 25 mph
[14], a 66.67% speed enhancement. Relatively speaking, the
size of the entire region shrinks by 64% with more dense
deployment of checkpoints. Compared with the results in
Fig. 2 (c), a performance enhancement up to 58% can be
seen. Respectively, Fig 5 (c) shows the results until the
global view forms at the seed(s) (with both Alg. 3 and
Alg. 4) with the same vehicle speedup. Compared with the
results in Fig. 3 (c), an enhancement up to 57% can be seen.
Next, by allowing in-out traffic of the interaction, the
above closed system will be transformed to an open system. Each checkpoint will apply Alg. 5. Initiated from the
same seeds (in the closed system), the counting will reach
the “complete status” at the global view level. To fetch
such a “complete status” and to verify the correctness of
our approach, each checkpoint will apply Alg. 4 to those
non-interaction traffic (enclosed by interaction as a closed
system) after its local counting stabilizes in Alg. 5.
In Fig. 4 (a), the average time for Alg. 5 to converge is
demonstrated. Fig. 4 (b) shows the performance of the same
procedure after the same speedup in Fig. 4 (c). Compared
with the results in Fig. 4 (a), a 30 ∼ 40% enhancement
is achieved. Respectively, Fig. 5 (a) and Fig. 5 (b), one
with the limit of 15 mph and the other with the limit of
25 mph, demonstrate the total time needed after both Alg. 5
and Alg. 4 converge.
We summarize our observations as follows.
1) The experimental results prove that our counting in
both closed and open systems does not have any
mis- or double-counting. We also verify the “complete
status” after Alg. 5 stabilizes in the open system.
2) The speed of achieving the “complete status” in the
open system mainly relies on the convergence speed
of counting on non-interaction traffic, which is similar
to counting in the closed system. Thus, the difference
between Fig. 2 (c) and Fig. 4 (a) is limited.
3) We noticed the impact of the traffic pattern. Once
labeled in the closed system, a vehicle will keep
its leading role in the frontier wave and deliver the
information without any delay. In the open system,
when such a vehicle joins the interaction traffic, the
border intersection must wait for another vehicle to
resume the information relay, causing a considerable
delay when many vehicles are through traffic in New
York City. Thus, the performance shown in Fig. 4 (a)
and Fig. 5 (a) are lower than those in Fig. 2 (c) and
Fig. 3 (c), respectively. But such a difference is in a
very limited range.
4) The speed of our counting mainly relies on how
quickly the spanning tree with the predecessor-andsuccessor relation can be built in information constitution as well as in information collection. The average
vehicle velocity (or the relative size of counting region) is key. A proportional enhancement can be seen
20
10
0
2
30
20
10
0
20
4
elapsed time (min)
elapsed time (min)
elapsed time (min)
30
2
40
6
10
100
8
100
40
20
0
2
# of vehicles (% of daily traffic)
10
100
80
# of vehicles (% of daily traffic)
(c) average
60
40
20
0
20
4
2
40
6
60
8
# of initial seeds
80
10
40
6
elapsed time (min)
elapsed time (min)
elapsed time (min)
20
4
Elapsed time (9 ∼ 30 min) of Alg. 3 in the closed system of New York midtown.
60
40
20
0
2
10
100
40
6
60
8
# of initial seeds
(a) maximum
20
4
40
6
80
# of vehicles (% of daily traffic)
60
20
4
60
100
2
(b) minimum
Figure 2.
10
0
60
# of initial seeds
(a) maximum
8
# of initial seeds
10
40
6
80
# of vehicles (% of daily traffic)
20
20
4
60
8
# of initial seeds
30
60
8
# of initial seeds
80
# of vehicles (% of daily traffic)
10
100
80
# of vehicles (% of daily traffic)
(c) average
(b) minimum
40
20
0
2
60
40
20
0
20
4
elapsed time (min)
60
elapsed time (min)
elapsed time (min)
Figure 3. Time (20 ∼ 50 min) needed for the seed(s) to obtain the global view after both Alg. 3 and Alg. 4 converge in the closed system of New York
midtown.
2
40
6
8
4
100
# of initial seeds
(a) time to reach a “complete status”
0
2
20
4
40
6
60
8
# of vehicles (% of daily traffic)
10
40
6
80
10
20
20
60
# of initial seeds
30
8
80
10
100
60
# of initial seeds
# of vehicles (% of daily traffic)
10
100
80
# of vehicles (% of daily traffic)
(c) speedup in the closed system for comparison
(b) time after speed limit lifted to 25 mph
40
20
0
2
20
4
40
6
60
8
# of initial seeds
10
100
80
# of vehicles (% of daily traffic)
(a) time for seed(s) to fetch a “complete status”
60
elapsed time (min)
60
elapsed time (min)
elapsed time (min)
Figure 4. (a) Elapsed time (on average) of Alg. 5 in the open system of New York midtown, (b) performance of Alg. 5 (34 ∼ 40% quicker vs. Fig. 4 (a))
after vehicle speed increased by 66%, and (c) performance of Alg. 3 (the version of our counting in the closed system) after the same speed enhancement
(up to 58% quicker vs. Fig. 2 (c)).
40
20
0
2
20
4
40
6
60
8
# of initial seeds
10
100
80
# of vehicles (% of daily traffic)
(b) time after speed limit lifted to 25 mph
30
20
10
0
2
20
4
40
6
8
# of initial seeds
60
10
100
80
# of vehicles (% of daily traffic)
(c) speedup in the closed system for comparison
Figure 5. (a) Elapsed time (on average) for both Alg. 5 and Alg. 4 to converge in the open system of New York midtown, (b) performance (34 ∼ 40%
quicker vs. Fig. 5 (a)) after vehicle speed increased by 66%, and (c) performance of Alg. 3 and Alg. 4 (the version of counting & collection in the closed
system) after the same speed enhancement (up to 57% quicker vs. Fig. 3 (c)).
in Fig. 4 (b) and Fig. 5 (b) from the open system,
and Fig. 4 (c) and Fig. 5 (c) from the closed system.
Those results also demonstrate the scalability of our
approach as a solution for the large-scale problem.
5) The time needed in our information constitution (in
Fig. 2 and Fig. 4) and information collection (in
Fig. 3 and Fig. 5) are in proportion to how quickly
a vehicle travels along the diameter of our target
region. The counting spreads as a wave. The impact of
other factors such as overtake, traffic block, one-way
street, lane change, and lossy communication can be
mitigated. Due to the high traffic volume, we have
enough vehicle appearance to end the local counting
and do not need police patrol.
6) The speedup of our counting by increasing the number
of seeds is not significant, until the spanning trees
initiated by each seed can evenly cover the entire
target region. This raises a concern of cost in the
collection of the “complete status” in the open system,
when each checkpoint along the border is deployed as
a global data sink. The delay needed to collect the
global snapshot from the border to the seed is still
considerable, while such a costly deployment itself
cannot significantly speed up our counting process.
Our results suggest the cost-effective deployment with
only one single sink.
VI. C ONCLUSION
In this paper, an infrastructure-less scheme is provided
for precisely counting all vehicles in a target region without
any mis- or double-counting. The scheme is applicable for
both closed and open road systems. The unique directive
is to provide a synchronization without any disruption on
vehicle trajectory or support of global infrastructure in order
to mitigate the impact of inconsistent local views among
different checkpoints. Our approach is implemented under
an “everyone” model in a fully-distributed manner, in order
to achieve the reliability and scalability. With the counting
result, we can have better strategies in resource management. For instance, our result can be used to determine an
appropriate level in disaster evacuation, or to evaluate the
market need for a new car service. In our future work, we
will conduct further studies on the impact of traffic patterns,
so that even better solutions can be achieved.
ACKNOWLEDGMENT
This research work was supported in part by NSF
grants ECCS 1231461, ECCS 1128209, CNS 1138963, CNS
1065444, and CCF 1028167.
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